Our universe enjoys a lot of symmetry. On the largest scales, the universe is homogeneous and isotropic, consistent with a spatially-flat Friedmann-Robertson-Walker space-time. The density inhomogeneities in the early universe, which later give rise to galaxies and the large scale structure, are the simplest imaginable: their statistics are approximately scale invariant and well-described by a gaussian random field. In this talk, I will describe three broad classes of early universe theories that can account for these observations: single-field inflation, multi-field inflation and the recently-proposed pseudo-conformal universe. I will show that each class is associated with distinct symmetries, which imply subtle constraints on the form of the primordial density correlation functions.